Karlovych, OleksiyShargorodsky, Eugene2025-08-292025-08-292025-11-301405-213XPURE: 103978609PURE UUID: 18465c62-bb29-498c-a9fb-334c978b40e9Scopus: 85209822216WOS: 001351491800001http://hdl.handle.net/10362/187197Funding Information: Open access funding provided by FCT|FCCN (b-on). This work is funded by national funds through the FCT - Fundação para a Ciência e a Tecnologia, I.P., under the scope of the projects UIDB/00297/2020 (https://doi.org/10.54499/UIDB/00297/2020) and UIDP/ 00297/2020 (https://doi.org/10.54499/UIDP/00297/2020) (Center for Mathematics and Applications). Publisher Copyright: © The Author(s) 2024.Let X be a Banach function space over the unit circle such that the Riesz projection P is bounded on X and let H[X] be the abstract Hardy space built upon X. We show that the essential norm of the Toeplitz operator T(a):H[X]→H[X] coincides with ‖a‖L∞ for every a∈C+H∞ if and only if the essential norm of the backward shift operator T(e-1):H[X]→H[X] is equal to one, where e-1(z)=z-1. This result extends an observation by Böttcher, Krupnik, and Silbermann for the case of classical Hardy spaces.9344811engAbstract Hardy spaceBanach function spaceEssential normToeplitz operatorGeneral Mathematicsjournal article10.1007/s40590-024-00689-2https://www.scopus.com/pages/publications/85209822216https://www.webofscience.com/wos/woscc/full-record/WOS:001351491800001